1
Chapter 1

Congruence, Proof, and Constructions

This chapter begins by precisely defining some geometric objects such as angles, circles, parallel lines, perpendicular lines, and line segments. Here, students need to pay attention to details to define each object correctly. In addition, except for the line segments definition, a definition in terms of transformations is presented for each of the other geometric objects. Later, transformations are defined as functions that can be applied to geometric figures. Also, it is studied when a transformation is a rigid motion as well as the composition of transformations. Having these definitions set allows the student to work with the definition of symmetry and explore whether a plane figure has some sort of symmetry - rotational symmetry or line symmetry. Finally, these two last types of symmetries are investigated through interactive applets where different transformations can be applied to a given polygon.

Next, a formal definition of rotations, translations, and reflections of plane figures are presented after having explored some properties of each of the transformations via interactive applets. Additionally, students are taught how to perform each of these transformations by hand. Students investigate the relationship between the input and output for each transformation when applied to points in the coordinate plane. Also, compositions involving either two rotations, or two translations, or two reflections, or a translation and a reflection are performed to investigate which of them can be seen as a single transformation.

The following lessons focus on developing theorems about triangle congruence, lines and angles, and parallelograms. The proofs of several of these theorems are developed using transformations. First, the definition of congruent figures in terms of rigid motions is presented. Then, congruent triangles are properly defined, and different criteria for proving triangle congruence are explored and established. The congruence theorems presented include the Side-Angle-Side Theorem, the Angle-Side-Angle Theorem, the Side-Side-Side Theorem, and the Angle-Angle-Side Theorem. Additionally, the Hypotenuse-Leg Congruence Theorem for right triangles is given. Leaving triangle congruence behind, the next lesson focuses on establishing relationships between the angles formed when a transversal cuts two parallel lines. These relationships are first investigated using interactive applets and then are formally stated and proven. The resulting theorems introduced include the Vertical Angles Theorem, the Corresponding Angles Theorem, the Alternate Interior Angles Theorem, the Alternate Exterior Angles Theorem, and the Perpendicular Bisector Theorem.

The next two lessons are fully devoted to investigating different properties of triangles, like the fact that, no matter the type of triangle, the sum of the interior angles is always 180 degrees. The definitions of centroid, circumcenter, incenter, and orthocenter are given along with a theorem highlighting their property. Relationships between the base angles of an isosceles triangle are set as well as the relationship between the midsegment of a triangle and its third side. Finally, it is mentioned that the centroid, the circumcenter, and the orthocenter of a triangle are collinear, and the line passing through them is called a Euler's Line. To close this group of theorem lessons, parallelograms are studied with the aim of discovering their different properties. Starting with general parallelograms, relationships between opposite sides, opposite angles, and diagonals are investigated using interactive applets. Additionally, criteria for determining when a quadrilateral is a parallelogram are presented. Similarly, criteria for determining whether a parallelogram is a rectangle or a rhombus are shown.

Before finishing the chapter and motivated through different applets, students learn how to perform different geometric constructions using different tools like a compass, a straightedge, a piece of string, or folding paper. The constructions include copying angles and segments, drawing parallel or perpendicular lines to a given line, drawing an angle bisector, and drawing a perpendicular bisector. The chapter finishes by showing in detail how to construct regular polygons inscribed in a circle - equilateral triangles, squares, regular pentagons, and regular hexagons. Once a definition or theorem is taught, it is put into practice by solving real-world situations.

2
Chapter 2

Similarity, Proof, and Trigonometry

The first set of lessons of this chapter is focused on the study of dilations and similarity transformations. First, the definition of a dilation and its scale factor is presented. Then, using interactive applets, properties of dilations - the dilation of a line is a line, dilations preserve angle measures - are investigated. Once the definition and properties have been set, dilations are applied to different objects like polygons, figures in the coordinate plane, or general images. Students also learn how to perform a dilation using a compass.

Later, similarity transformations are introduced as well as the definition of similar figures. Since similarity transformations are based on dilations, they share the same properties. However, this lesson shows some criteria for determining whether two polygons are similar. Furthermore, triangles come to play, and theorems like the Angle-Angle Similarity Theorem, the Side-Side-Side Similarity Theorem, and the Side-Angle-Side Similarity Theorem are developed. Apart from studying the similarity between two given triangles, some properties within a single triangle are studied and then formalized as theorems - the Triangle Proportionality Theorem and its converse, Right Triangle Proportionality Theorem, Geometric Mean Altitude Theorem, and Geometric Mean Leg Theorem. Additionally, the Pythagorean Theorem is proven using similarity. This set of lessons finishes by investigating different properties of quadrilaterals, such as the fact that the midpoints of the sides of any quadrilateral form a parallelogram. In addition, the Triangle Angle Bisector Theorem and its converse are presented and put into practice.

Next, similarity is applied in right triangles to understand trigonometric ratios. Here, the sine, cosine, and tangent ratios for acute angles are introduced as well as their corresponding inverse and reciprocal ratios. Each of these ratios is put into action to find a missing length in a given triangle. Furthermore, it is shown that the sine and cosine of complementary angles are equal, which helps to rewrite expressions using sine in terms of cosine and vice-versa. Later, the angle of elevation and depression are introduced and used to model different real-world scenarios where trigonometric ratios can be applied. Finally, trigonometric ratios are used to find the volume and density of different objects.

To finish the chapter, three equivalent formulas for finding the area of a triangle using the sine ratio are shown and put into practice. Also, the Law of Sines and the Law of Cosines are stated, proven, and used to find missing measures in a triangle modeling different real-world scenarios.

3
Chapter 3

Extending to Three Dimensions

In this chapter, students will use ratios to compare the areas and volumes of similar figures. Given the scale factor between two similar figures, students will be able to express the area and volume scale factors and calculate the missing measures of the figures. Students will first examine circles, developing the formula for the circumference of a circle using similarity and the area formula using dissection arguments.

In relation to circles, cylinders and their properties will be examined. By the use of Cavalieri's Principle, students will be able to justify why the volume formula for cylinders works. Students will also develop the formula for the surface area of a cylinder. In addition to cylinders, some other common three-dimensional figures - cones, pyramids, and spheres - and their properties will be discussed throughout the chapter. The volume formulas for cones, pyramids, and spheres will be developed by Cavalieri's Principle and dissection arguments. The surface area formulas will be developed considering the nets of the solids. Furthermore, students are also expected to explore the relationship between the volumes of cylinders, cones, and spheres, given that they have the same linear measures.

After exploring the properties of the common solids, their cross-sections and the solids generated by rotations of two-dimensional objects will be identified and analyzed. Finally, real-life situations will be modeled using geometric and algebraic reasoning. Some of the examples require the derivation of expressions to represent areas of varying cross-sections, the use of Cavalieri's principle, and the conversion factor, while others require applying the concepts of mass, volume, and density.

4
Chapter 4

Connecting Algebra and Geometry Through Coordinates

In this unit students will learn about the Distance Formula and its corresponding proof. They will use this formula to find the distance between two points in the Cartesian plane, to prove a quadrilateral is a rectangle (in conjunction with the converse of the Pythagorean theorem), and to determine whether a point whose coordinates are given lies on a circle whose equation is to be found. Students will also learn the definition of a midpoint and the Midpoint Formula, along with its corresponding proof. They will use this formula to find the midpoint between two points in the Cartesian plane and to prove properties of a parallelogram. Moreover, students will use rigid motions to prove theorems using the Distance and the Midpoint formulas in the Cartesian plane.

Students will also investigate and learn the relationship between the slopes of parallel lines in a coordinate plane. The Slopes of Parallel Lines Theorem will be discussed, proved, and used to find the slope of a parallel line to a given line, which may or may not be written in slope-intercept form. The graph of a line and a point not on the line will be given, and students will be asked to find the equation of the parallel line through the point. Systems of equations with parallel or overlapping lines will be discussed, as well as properties of different systems of equations. Properties of parallel lines will be used to classify parallelograms. Furthermore, slope triangles of different lines will be investigated, and students will learn about the relationship between the slopes of perpendicular lines in a coordinate plane. The Slopes of Perpendicular Lines Theorem will be discussed, proved, and used to find the slope of a perpendicular line to a given line, which may or may not be written in slope-intercept form. The equation of a perpendicular line to a given line will be investigated, and a perpendicular line to a line whose graph is given, through a given point, will be found.

This unit will also discuss positions of points in a line segment, the partitioning of a directed line segment, and a general method for partitioning directed line segments. This will be used to find points on a triangle's perimeter and to partition distances in real life examples.

Using the Distance formula and the Pythagorean theorem, students will work with the perimeter and area of different polygons in the Cartesian plane. They will solve real-life problems and investigate properties of polygons and compound shapes.

Finally, the definition of a parabola will be given. Students will investigate points that are equidistant from a pair of given options and properties of parabolas, and will learn about the focus and directrix of a parabola. They will use the Distance Formula to calculate points on a parabola and will write its equation, given the focus and directrix. Vertical and horizontal parabolas will be discussed, as well as the use and applications of parabolas in the real world.

5
Chapter 5

Circles With and Without Coordinates

In this unit, students will interact with concentric circles to discover that they are similar through dilation. Students will find scale factors that correspond to enlargements and to reductions. They will also interact with non-concentric circles to realize that non-concentric circles are also similar through translation and dilation. They will find the transformation or sequence of transformations that maps one circle onto another. In conclusion, it will be shown that all circles are similar through similarity transformations.

Inscribed and central angles of a circle will be defined, as well as their intercepted arc. Students will interact with inscribed angles to find that the measure of an inscribed angle whose intercepted arc is a semicircle is 90 degrees. Students will also interact with inscribed and central angles, learning that if they intercept the same arc, then the measure of the central angle is twice the measure of the inscribed angle. Moreover, the tangent line to a circle will be defined and students will be able to use interactive graphs to realize that a tangent line to a circle is always perpendicular to the radius at the point of tangency. Furthermore, circumscribed angles will be defined and students will have the opportunity to explore them through interactive diagrams. Several theorems will be stated, proved, and used to find missing measures.

  • The Inscribed Angle Theorem
  • The Inscribed Angles of a Circle Theorem
  • The Tangent to a Circle Theorem
  • The Circumscribed Angle Theorem

Inscribed and circumscribed circles of a triangle will be defined and students will have the opportunity of investigating them through interactions. The incenter, circumcenter, and centroid of a triangle will also be defined, including a recollection of the concepts of angle bisector, perpendicular bisector, and median. Their properties will be used to solve problems. Students will learn the method for drawing both the inscribed and circumscribed circles of a triangle, and will use them to solve real-life problems.

Students will investigate and interact with inscribed quadrilaterals and realize that opposite angles are supplementary. Inscribed quadrilaterals will be defined along with a mention of the alternative name, cyclic quadrilateral. Students will see the formal proof that opposite angles in an inscribed quadrilateral are supplementary, and use this fact to find missing angle measures. The Cyclic Quadrilateral Exterior Angle Theorem will be stated, proved, and used to find missing angle measures. Students will also learn that the perpendicular bisectors of a cyclic quadrilateral are concurrent.

Recalling the definition of a tangent line to a circle, students will learn how to draw the tangent through an outer point, and use this construction to solve problems. The relationship between tangents and lines of symmetry will be investigated. The External Tangent Congruence Theorem will be stated, proved, and used. Applications of tangent lines to circles in real life will be seen.

This unit will also explore the relationship between radian measure and arc length. Students will learn about radians and how to convert from radians to degrees and vice versa. They will calculate radian measures and compare radian measures of concentric circles. Moreover, radians will be used to calculate arc lengths and angle measures.

In addition to their work with arc lengths, students will also work with the area of circle sectors. The definition of a circle sector will be stated and the formula for the area of a sector will be proved. Students will use this formula to solve sector problems.

In the Cartesian coordinate system, students will use the Distance Formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, students will identify the radius and the center, and will draw the graph in the coordinate plane. Students will algebraically verify whether a point is on a circle, and write the standard equation of a circle by completing squares.

Coordinates will be used to write proofs. Properties of parallelograms, circles, and triangles will be proved by using coordinates. Moreover, real life problems involving these geometric shapes will be solved by using coordinates.

Finally, the unit will cover the modeling of real life objects with circles. Properties of circles, radius, and diameter will be used to calculate area and circumference of ordinary objects.

6
Chapter 6

Applications of Probability

In this chapter, students will study the fundamental concepts and rules of probability. In the beginning, students will be introduced to several definitions: an event and its complement, an outcome and sample space, and a probability experiment and its elements. From there, the difference between experimental and theoretical probability will be explained. Unions and intersections of two or more events will be taught with the usage of Venn diagrams. Students will learn that the occurrence of some events can affect the occurrence of other events, and investigating that fact will help them discover the difference between dependent and independent events. Students will be encouraged to explore the connection between independence and conditional probability, then analyze a variety of conditional situations modeled by tree diagrams, frequency tables, and Venn diagrams. Numerous real-life situations will be presented and analyzed from the perspective of independence and conditional probability. One of the lessons is deeply focused on geometric probability problems in one-, two-, and three-dimensional situations. Students will also learn to interpret data given in two-way frequency tables and to calculate joint, marginal, and conditional relative frequencies. Students will practice using the Addition and Multiplication Rules of Probabilities when calculating the probabilities of compound events. The concepts of permutations, combinations, and simulations will be introduced and connected to the computation of probabilities. Students will be taught how calculating probabilities of events helps to choose between different decision-making methods and whether or not decisions made using those methods are fair. Finally, strategizing and decision-making principles will be explored and put into practice through different concepts of probability.